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Chapter 16

Macroeconomics Toolkit

In this chapter, we present the key tools used in the macroeconomics part of thistextbook. This toolkit serves two main functions:

1. Because these tools appear in multiple chapters, the toolkit serves as areference. When using a tool in one chapter, you can refer back to thetoolkit to find a more concise description of the tool as well as links toother parts of the book where the tool is used.

2. You can use the toolkit as a study guide. Once you have workedthrough the material in the chapters, you can review the tools usingthis toolkit.

The chart below shows the main uses of each tool in green, and the secondary usesare in orange.

773

16.1 The Labor Market

The labor market is the market in which labor services are traded. Individual laborsupply comes from the choices of individuals or households about how to allocatetheir time. As the real wage (the nominal wage divided by the price level)increases, households supply more hours to the market, and more householdsdecide to participate in the labor market. Thus the quantity of labor suppliedincreases. The labor supply curve of a household is shifted by changes in wealth. Awealthier household supplies less labor at a given real wage.

Labor demand comes from firms. As the real wage increases, the marginal cost ofhiring more labor increases, so each firm demands fewer hours of labor input—thatis, a firm’s labor demand curve is downward sloping. The labor demand curve of afirm is shifted by changes in productivity. If labor becomes more productive, thenthe labor demand curve of a firm shifts rightward: the quantity of labor demandedis higher at a given real wage.

The labor market equilibrium is shown in Figure 16.1 "Labor Market Equilibrium".The real wage and the equilibrium quantity of labor traded are determined by theintersection of labor supply and labor demand. At the equilibrium real wage, thequantity of labor supplied equals the quantity of labor demanded.

Chapter 16 Macroeconomics Toolkit

774

Figure 16.1 Labor Market Equilibrium

Key Insights

• Labor supply and labor demand depend on the real wage.• Labor supply is upward sloping: as the real wage increases, households

supply more hours to the market.• Labor demand is downward sloping: as the real wage increases, firms

demand fewer hours of work.• A market equilibrium is a real wage and a quantity of hours such that

the quantity demanded equals the quantity supplied.

The Main Uses of This Tool

• Chapter 4 "The Interconnected Economy"• Chapter 5 "Globalization and Competitiveness"• Chapter 8 "Jobs in the Macroeconomy"• Chapter 11 "Inflations Big and Small"• Chapter 12 "Income Taxes"• Chapter 14 "Balancing the Budget"


16.1 The Labor Market 775

16.2 Choices over Time

Individuals make decisions that unfold over time. Because individuals choose howto spend income earned over many periods on consumption goods over manyperiods, they sometimes wish to save or borrow rather than spend all their incomein every period.

Figure 16.2 "Choices over Time" shows examples of these choices over a two-yearhorizon. The individual earns income this year and next. The combinations ofconsumption that are affordable and that exhaust all of an individual’s income areshown on the budget line, which in this case is called an intertemporal budgetconstraint. The slope of the budget line is equal to (1 + real interest rate), which isequivalent to the real interest factor. The slope is the amount of consumption thatcan be obtained tomorrow by giving up a unit of consumption today.

The preferred point is also indicated; it is the combination of consumption this yearand consumption next year that the individual prefers to all the points on thebudget line. The individual in part (a) of Figure 16.2 "Choices over Time" isconsuming less this year than she is earning: she is saving. Next year she can useher savings to consume more than her income. The individual in part (b) of Figure16.2 "Choices over Time" is consuming more this year than he is earning: he isborrowing. Next year, his consumption will be less than his income because he mustrepay the amount borrowed this year.

When the real interest rate increases, individuals will borrow less and (usually) savemore (the effect of interest rate changes on saving is unclear as a matter of theorybecause income effects and substitution effects act in opposite directions). Thusindividual loan supply slopes upward.

Of course, individuals live for many periods and make frequent decisions onconsumption and saving. The lifetime budget constraint is obtained using the ideaof discounted present value:

discounted present value of lifetime income = discounted present value of lifetimeconsumption.

The left side is a measure of all the disposable income the individual will receiveover his lifetime (disposable means after taking into account taxes paid to thegovernment and transfers received from the government). The right side calculatesthe value of consumption of all goods and services over an individual’s lifetime.


776

Key Insights

• Over a lifetime, an individual’s discounted present value ofconsumption will equal the discounted present value of income.

• Individuals can borrow or lend to obtain their preferred consumptionbundle over their lifetimes.

• The price of borrowing is the real interest rate.

Figure 16.2 Choices over Time

The Main Use of This Tool

• Chapter 13 "Social Security"


16.2 Choices over Time 777

16.3 Discounted Present Value

Discounted present value is a technique used to add dollar amounts over time. Weneed this technique because a dollar today has a different value from a dollar in thefuture.

The discounted present value this year of $1.00 that you will receive next year is asfollows:

If the nominal interest rate is 10 percent, then the nominal interest factor is 1.1,so $1 next year is worth $1/1.1 = $0.91 this year. As the interest rate increases, thediscounted present value decreases.

More generally, we can compute the value of an asset this year from the followingformula:

The flow benefit depends on the asset. For a bond, the flow benefit is a couponpayment. For a stock, the flow benefit is a dividend payment. For a fruit tree, theflow benefit is the yield of a crop.

If an asset (such as a bond) yields a payment next year of $10 and has a price nextyear of $90, then the “flow benefit from asset + price of the asset next year” is $100.

The value of the asset this year is then $100nominal interest factor .If the nominal interest

rate is 20 percent, then the value of the asset is $100/1.2 = 83.33.

We discount nominal flows using a nominal interest factor. We discount real flows(that is, flows already corrected for inflation) using a real interest factor, which isequal to (1 + real interest rate).

$1nominal interest factor

=$1

1 + nominal interest rate.

value of asset this year =flow benefit from asset this year + price of asset next year

nominal interest factor.


778

Key Insights

• If the interest rate is positive, then the discounted present value is lessthan the direct sum of flows.

• If the interest rate increases, the discounted present value willdecrease.

More Formally

Denote the dividend on an asset in period t as Dt. Define Rt as the cumulative effect of

interest rates up to period t. For example, R2 = (1 + r1)(1 + r2). Then the value of an

asset that yields Dt dollars in every year up to year T is given by

If the interest rate is constant (equal to r), then the one period interest factor is R =1 + r, and Rt = Rt.

The discounted present value tool is illustrated in Table 16.1 "Discounted PresentValue with Different Interest Rates". The number of years (T) is set equal to 5. Thetable gives the value of the dividends in each year and computes the discountedpresent values for two different interest rates. For this example, the annual interestrates are constant over time.

Table 16.1 Discounted Present Value with Different Interest Rates

YearDividend

($)Discounted Present Value

with R = 1.05 ($)Discounted Present Value

with R = 1.10 ($)

1 100 100 100

2 100 95.24 90.91

3 90 81.63 74.38

4 120 103.66 90.16

5 400 329.08 273.20

Discountedpresent value

709.61 628.65

asset value =D1

R1+

D2

R2+

D3

R3+ ... +

DT

RT.


16.3 Discounted Present Value 779


• Chapter 8 "Jobs in the Macroeconomy"• Chapter 10 "Understanding the Fed"• Chapter 13 "Social Security"• Chapter 14 "Balancing the Budget"• Chapter 15 "The Global Financial Crisis"


16.3 Discounted Present Value 780

16.4 The Credit (Loan) Market (Macro)

Consider a simple example of a loan. Imagine you go to your bank to inquire about aloan of $1,000, to be repaid in one year’s time. A loan is a contract that specifiesthree things:

1. The amount being borrowed (in this example, $1,000)2. The date(s) at which repayment must be made (in this example, one

year from now)3. The amount that must be repaid

What determines the amount of the repayment? The lender—the bank—is asupplier of credit, and the borrower—you—is a demander of credit. We use theterms credit and loans interchangeably. The higher the repayment amount, the moreattractive this loan contract will look to the bank. Conversely, the lower therepayment amount, the more attractive this contract is to you.

If there are lots of banks that are willing to supply such loans, and lots of people likeyou who demand such loans, then we can draw supply and demand curves in thecredit (loan) market. The equilibrium price of this loan is the interest rate at whichsupply equals demand.

In macroeconomics, we look at not only individual markets like this but also thecredit (loan) market for an entire economy. This market brings together suppliersof loans, such as households that are saving, and demanders of loans, such asbusinesses and households that need to borrow. The real interest rate is the“price” that brings demand and supply into balance.

The supply of loans in the domestic loans market comes from three differentsources:

1. The private saving of households and firms2. The saving of governments (in the case of a government surplus)3. The saving of foreigners (when there is a flow of capital into the

domestic economy)

Households will generally respond to an increase in the real interest rate byreducing current consumption relative to future consumption. Households that aresaving will save more; households that are borrowing will borrow less. Higher


781

interest rates also encourage foreigners to send funds to the domestic economy.Government saving or borrowing is little affected by interest rates.

The demand for loans comes from three different sources:

1. The borrowing of households and firms to finance purchases, such ashousing, durable goods, and investment goods

2. The borrowing of governments (in the case of a government deficit)3. The borrowing of foreigners (when there is a flow of capital from the

domestic economy)

As the real interest rate increases, investment and durable goods spendingdecrease. For firms, a high interest rate represents a high cost of fundinginvestment expenditures. This is an application of discounted present value and isevident if a firm borrows to purchase capital. It is also true if it uses internal funds(retained earnings) to finance investment because the firm could always put thosefunds into an interest-bearing asset instead. For households, higher interest rateslikewise make it more costly to borrow to purchase housing and durable goods. Thedemand for credit decreases as the interest rate rises. When it is expensive toborrow, households and firms will borrow less.

Equilibrium in the market for loans is shown in Figure 16.3 "The Credit Market". Onthe horizontal axis is the total quantity of loans in equilibrium. The demand curvefor loans is downward sloping, whereas the supply curve has a positive slope. Loanmarket equilibrium occurs at the real interest rate where the quantity of loanssupplied equals the quantity of loans demanded. At this equilibrium real interestrate, lenders lend as much as they wish, and borrowers can borrow as much as theywish. Equilibrium in the aggregate credit market is what ensures the balance offlows into and out of the financial sector in the circular flow diagram.

Key Insights

• As the real interest rate increases, more loans are supplied, and fewerloans are demanded.

• Adjustment of the real interest rate ensures that, in the circular flowdiagram, the flows into the financial sector equal the flows from thesector.


• Chapter 4 "The Interconnected Economy"


16.4 The Credit (Loan) Market (Macro) 782

• Chapter 9 "Money: A User’s Guide"• Chapter 10 "Understanding the Fed"• Chapter 11 "Inflations Big and Small"• Chapter 14 "Balancing the Budget"

Figure 16.3 The Credit Market


16.4 The Credit (Loan) Market (Macro) 783

16.5 Correcting for Inflation

If you have some data expressed in nominal terms (for example, in dollars), and youwant to convert them to real terms, you should use the following four steps.

1. Select your deflator. In most cases, the Consumer Price Index (CPI) isthe best deflator to use. You can find data on the CPI (for the UnitedStates) at the Bureau of Labor Statistics website (http://www.bls.gov).

2. Select your base year. Find the value of the index in that base year.3. For all years (including the base year), divide the value of the index in

that year by the value in the base year. The value for the base year is 1.4. For each year, divide the value in the nominal data series by the

number you calculated in step 3. This gives you the value in “base yeardollars.”

Table 16.2 "Correcting Nominal Sales for Inflation" shows an example. We have dataon the CPI for three years, as listed in the second column. The price index is createdusing the year 2000 as a base year, following steps 1–3. Sales measured in millions ofdollars are given in the fourth column. To correct for inflation, we divide sales ineach year by the value of the price index for that year. The results are shown in thefifth column. Because there was inflation each year (the price index is increasingover time), real sales do not increase as rapidly as nominal sales.

Table 16.2 Correcting Nominal Sales for Inflation

Year CPIPrice Index (2000

Base)Sales

(Millions)Real Sales (Millions of Year 2000

Dollars)

2000 172.2 1.0 21.0 21.0

2001 177.1 1.03 22.3 21.7

2002 179.9 1.04 22.9 21.9

Source: Bureau of Labor Statistics for the Consumer Price Index

This calculation uses the CPI, which is an example of a price index. To see how aprice index like the CPI is constructed, consider Table 16.3 "Constructing a PriceIndex", which shows a very simple economy with three goods: T-shirts, musicdownloads, and meals. The prices and quantities purchased in the economy in 2012and 2013 are summarized in the table.


784

http://www.bls.gov

Table 16.3 Constructing a Price Index

Year T-shirtsMusic

DownloadsMeals

Cost of 2013Basket

PriceIndex

Price($)

QuantityPrice($)

QuantityPrice($)

Quantity Price ($)

2012 20 10 1 50 25 6 425 1.00

2013 22 12 0.80 60 26 5 442 1.04

To construct a price index, you must choose a fixed basket of goods. For example,we could use the goods purchased in 2013 (12 T-shirts, 60 downloads, and 5 meals).This fixed basket is then priced in different years. To construct the cost of the 2013basket at 2013 prices, the product of the price and the quantity purchased for eachgood in 2013 is added together. The basket costs $442. Then we calculate the cost ofthe 2013 basket at 2012 prices: that is, we use the prices of each good in 2012 andthe quantities purchased in 2013. The sum is $425. The price index is constructedusing 2012 as a base year. The value of the price index for 2013 is the cost of thebasket in 2013 divided by its cost in the base year (2012).

When the price index is based on a bundle of goods that represents total output inan economy, it is called the price level. The CPI and gross domestic product (GDP)deflator are examples of measures of the price level (they differ in terms of exactlywhich goods are included in the bundle). The growth rate of the price level (itspercentage change from one year to the next) is called the inflation rate.

We also correct interest rates for inflation. The interest rates you typically seequoted are in nominal terms: they tell you how many dollars you will have to repayfor each dollar you borrow. This is called a nominal interest rate. The realinterest rate tells you how much you will get next year, in terms of goods andservices, if you give up a unit of goods and services this year. To correct interestrates for inflation, we use the Fisher equation:

real interest rate ≈ nominal interest rate − inflation rate.

For more details, see Section 16.14 "The Fisher Equation: Nominal and Real InterestRates" on the Fisher equation.

Key Insights

• Divide nominal values by the price index to create real values.


16.5 Correcting for Inflation 785

• Create the price index by calculating the cost of buying a fixed basketin different years.


• Chapter 3 "The State of the Economy"• Chapter 4 "The Interconnected Economy"• Chapter 5 "Globalization and Competitiveness"• Chapter 9 "Money: A User’s Guide"• Chapter 11 "Inflations Big and Small"• Chapter 12 "Income Taxes"


16.5 Correcting for Inflation 786

16.6 Supply and Demand

The supply-and-demand framework is the most fundamental framework ineconomics. It explains both the price of a good or a service and the quantityproduced and purchased.

The market supply curve comes from adding together the individual supplycurves of firms in a particular market. A competitive firm, taking prices as given,will produce at a level such that

price = marginal cost.

Marginal cost usually increases as a firm produces more output. Thus an increase inthe price of a product creates an incentive for firms to produce more—that is, thesupply curve of a firm is upward sloping. The market supply curve slopes upward aswell: if the price increases, all firms in a market will produce more output, andsome new firms may also enter the market.

A firm’s supply curve shifts if there are changes in input prices or the state oftechnology. The market supply curve is shifted by changes in input prices andchanges in technology that affect a significant number of the firms in a market.

The market demand curve comes from adding together the individual demandcurves of all households in a particular market. Households, taking the prices of allgoods and services as given, distribute their income in a manner that makes themas well off as possible. This means that they choose a combination of goods andservices preferred to any other combination of goods and services they can afford.They choose each good or service such that

price = marginal valuation.

Marginal valuation usually decreases as a household consumes more of a product. Ifthe price of a good or a service decreases, a household will substitute away fromother goods and services and toward the product that has become cheaper—that is,the demand curve of a household is downward sloping. The market demand curveslopes downward as well: if the price decreases, all households will demand more.

The household demand curve shifts if there are changes in income, prices of othergoods and services, or tastes. The market demand curve is shifted by changes inthese factors that are common across a significant number of households.


787

A market equilibrium is a price and a quantity such that the quantity suppliedequals the quantity demanded at the equilibrium price (Figure 16.4 "MarketEquilibrium"). Because market supply is upward sloping and market demand isdownward sloping, there is a unique equilibrium price. We say we have acompetitive market if the following are true:

• The product being sold is homogeneous.• There are many households, each taking the price as given.• There are many firms, each taking the price as given.

A competitive market is typically characterized by an absence of barriers to entry,so new firms can readily enter the market if it is profitable, and existing firms caneasily leave the market if it is not profitable.

Key Insights

• Market supply is upward sloping: as the price increases, all firms willsupply more.

• Market demand is downward sloping: as the price increases, allhouseholds will demand less.

• A market equilibrium is a price and a quantity such that the quantitydemanded equals the quantity supplied.


16.6 Supply and Demand 788

Figure 16.4 Market Equilibrium

Figure 16.4 "Market Equilibrium" shows equilibrium in the market for chocolatebars. The equilibrium price is determined at the intersection of the market supplyand market demand curves.

More Formally

If we let p denote the price, qd the quantity demanded, and I the level of income,then the market demand curve is given by

qd = a − bp + cI,

where a, b, and c are constants. By the law of demand, b > 0. For a normal good, thequantity demanded increases with income: c > 0.

If we let qs denote the quantity supplied and t the level of technology, the marketsupply curve is given by

qs = d + ep + ft,



where d, e, and f are constants. Because the supply curve slopes upward, e > 0.Because the quantity supplied increases when technology improves, f > 0.

In equilibrium, the quantity supplied equals the quantity demanded. Set qs = qd = q*and set p = p* in both equations. The market clearing price (p*) and quantity (q*) areas follows:

and

q* = d + ep* + ft.


• Chapter 4 "The Interconnected Economy"• Chapter 9 "Money: A User’s Guide"

p* =(a + cI) − (d + f t)

b + e



16.7 Comparative Advantage

Comparative advantage explains why individuals and countries trade with eachother. Trade is at the heart of modern economies: individuals specialize inproduction and generalize in consumption. To consume many goods whileproducing relatively few, individuals must sell what they produce in exchange forthe output of others. Countries likewise specialize in certain goods and services andimport others. By so doing, they obtain gains from trade.

Table 16.4 "Hours of Labor Required" shows the productivity of two differentcountries in the production of two different goods. It shows the number of laborhours required to produce two goods—tomatoes and beer—in two countries:Guatemala and Mexico. From these data, Mexico has an absolute advantage in theproduction of both goods. Workers in Mexico are more productive at producingboth tomatoes and beer in comparison to workers in Guatemala.

Table 16.4 Hours of Labor Required

Tomatoes (1 Kilogram) Beer (1 Liter)

Guatemala 6 3

Mexico 2 2

In Guatemala, the opportunity cost of 1 kilogram of tomatoes is 2 liters of beer. Toproduce an extra kilogram of tomatoes in Guatemala, 6 hours of labor time must betaken away from beer production; 6 hours of labor time is the equivalent of 2 litersof beer. In Mexico, the opportunity cost of 1 kilogram of tomatoes is 1 liter of beer.Thus the opportunity cost of producing tomatoes is lower in Mexico than inGuatemala. This means that Mexico has a comparative advantage in the productionof tomatoes. By a similar logic, Guatemala has a comparative advantage in theproduction of beer.

Guatemala and Mexico can have higher levels of consumption of both beer andtomatoes if they trade rather than produce in isolation; each country shouldspecialize (either partially or completely) in the good in which it has a comparativeadvantage. It is never efficient to have both countries produce both goods.


791

Key Insights

• Comparative advantage helps predict the patterns of trade betweenindividuals and/or countries.

• A country has a comparative advantage in the production of a good ifthe opportunity cost of producing that good is lower in that country.

• Even if one country has an absolute advantage in all goods, it will stillgain from trading with another country.

• Although this example is cast in terms of countries, the same logic isalso used to explain production patterns between two individuals.


• Chapter 8 "Jobs in the Macroeconomy"


16.7 Comparative Advantage 792

16.8 Comparative Statics

Comparative statics is a tool used to predict the effects of exogenous variables onmarket outcomes. Exogenous variables shift either the market demand curve (forexample, news about the health effects of consuming a product) or the marketsupply curve (for example, weather effects on a crop). By market outcomes, wemean the equilibrium price and the equilibrium quantity in a market. Comparativestatics is a comparison of the market equilibrium before and after a change in anexogenous variable.

A comparative statics exercise consists of a sequence of five steps:

1. Begin at an equilibrium point where the quantity supplied equals thequantity demanded.

2. Based on a description of the event, determine whether the change inthe exogenous variable shifts the market supply curve or the marketdemand curve.

3. Determine the direction of this shift.4. After shifting the curve, find the new equilibrium point.5. Compare the new and old equilibrium points to predict how the

exogenous event affects the market.

Figure 16.5 "A Shift in the Demand Curve" and Figure 16.6 "A Shift in the SupplyCurve" show comparative statics in action in the market for Curtis Grandersonreplica shirts and the market for beer. In Figure 16.5 "A Shift in the Demand Curve",the market demand curve has shifted leftward. The consequence is that theequilibrium price and the equilibrium quantity both decrease. The demand curveshifts along a fixed supply curve. In Figure 16.6 "A Shift in the Supply Curve", themarket supply curve has shifted leftward. The consequence is that the equilibriumprice increases and the equilibrium quantity decreases. The supply curve shiftsalong a fixed demand curve.

Key Insights

• Comparative statics is used to determine the market outcome when themarket supply and demand curves are shifting.

• Comparative statics is a comparison of equilibrium points.• If the market demand curve shifts, then the new and old equilibrium

points lie on a fixed market supply curve.


793

• If the market supply curve shifts, then the new and old equilibriumpoints lie on a fixed market demand curve.

Figure 16.5 A Shift in the Demand Curve


16.8 Comparative Statics 794

Figure 16.6 A Shift in the Supply Curve


• Chapter 4 "The Interconnected Economy"• Chapter 7 "The Great Depression"• Chapter 11 "Inflations Big and Small"


16.8 Comparative Statics 795

16.9 Nash Equilibrium

A Nash equilibrium is used to predict the outcome of a game. By a game, we meanthe interaction of a few individuals, called players. Each player chooses an action andreceives a payoff that depends on the actions chosen by everyone in the game.

A Nash equilibrium is an action for each player that satisfies two conditions:

1. The action yields the highest payoff for that player given herpredictions about the other players’ actions.

2. The player’s predictions of others’ actions are correct.

Thus a Nash equilibrium has two dimensions. Players make decisions that are intheir own self-interests, and players make accurate predictions about the actions ofothers.

Consider the games in Table 16.5 "Prisoners’ Dilemma", Table 16.6 "Dictator Game",Table 16.7 "Ultimatum Game", and Table 16.8 "Coordination Game". The numbers inthe tables give the payoff to each player from the actions that can be taken, withthe payoff of the row player listed first.

Table 16.5 Prisoners’ Dilemma

Left Right

Up 5, 5 0, 10

Down 10, 0 2, 2

Table 16.6 Dictator Game

Number of dollars (x) 100 − x, x

Table 16.7 Ultimatum Game

Accept Reject

Number of dollars (x) 100 − x, x 0, 0


796

Table 16.8 Coordination Game

Left Right

Up 5, 5 0, 1

Down 1, 0 4, 4

• Prisoners’ dilemma. The row player chooses between the actionlabeled Up and the one labeled Down. The column player choosesbetween the action labeled Left and the one labeled Right. For example,if row chooses Up and column chooses Right, then the row player has apayoff of 0, and the column player has a payoff of 10. If the row playerpredicts that the column player will choose Left, then the row playershould choose Down (that is, down for the row player is her bestresponse to left by the column player). From the column player’sperspective, if he predicts that the row player will choose Up, then thecolumn player should choose Right. The Nash equilibrium occurs whenthe row player chooses Down and the column player chooses Right. Ourtwo conditions for a Nash equilibrium of making optimal choices andpredictions being right both hold.

• Social dilemma. This is a version of the prisoners’ dilemma in whichthere are a large number of players, all of whom face the same payoffs.

• Dictator game. The row player is called the dictator. She is given $100and is asked to choose how many dollars (x) to give to the columnplayer. Then the game ends. Because the column player does not movein this game, the dictator game is simple to analyze: if the dictator isinterested in maximizing her payoff, she should offer nothing (x = 0).

• Ultimatum game. This is like the dictator game except there is asecond stage. In the first stage, the row player is given $100 and told tochoose how much to give to the column player. In the second stage, thecolumn player accepts or rejects the offer. If the column player rejectsthe offer, neither player receives any money. The best choice of therow player is then to offer a penny (the smallest amount of moneythere is). The best choice of the column player is to accept. This is theNash equilibrium.

• Coordination game. The coordination game has two Nash equilibria. Ifthe column player plays Left, then the row player plays Up; if the rowplayer plays Up, then the column player plays Left. This is anequilibrium. But Down/Right is also a Nash equilibrium. Both playersprefer Up/Left, but it is possible to get stuck in a bad equilibrium.


16.9 Nash Equilibrium 797

Key Insights

• A Nash equilibrium is used to predict the outcome of games.• In real life, payoffs may be more complicated than these games

suggest. Players may be motivated by fairness or spite.

More Formally

We describe a game with three players (1, 2, 3), but the idea generalizesstraightforwardly to situations with any number of players. Each player chooses astrategy (s1, s2, s3). Suppose σ1(s1, s2, s3) is the payoff to player 1 if (s1, s2, s3) is the

list of strategies chosen by the players (and similarly for players 2 and 3). We put anasterisk (*) to denote the best strategy chosen by a player. Then a list of strategies(s*1, s*2, s*3) is a Nash equilibrium if the following statements are true:

σ1(s*1, s*2, s*3) ≥ σ1(s1, s*2, s*3)

σ2(s*1, s*2, s*3) ≥ σ2(s*1, s2, s*3)

σ3(s*1, s*2, s*3) ≥ σ3(s*1, s*2, s3)

In words, the first condition says that, given that players 2 and 3 are choosing theirbest strategies (s*2, s*3), then player 1 can do no better than to choose strategy s*1.

If a similar condition holds for every player, then we have a Nash equilibrium.


• Chapter 7 "The Great Depression"• Chapter 11 "Inflations Big and Small"• Chapter 15 "The Global Financial Crisis"


16.9 Nash Equilibrium 798

16.10 Foreign Exchange Market

A foreign exchange market is where one currency is traded for another. There is ademand for each currency and a supply of each currency. In these markets, onecurrency is bought using another. The price of one currency in terms of another(for example, how many dollars it costs to buy one Mexican peso) is called theexchange rate.

Foreign currencies are demanded by domestic households, firms, and governmentsthat wish to purchase goods, services, or financial assets denominated in thecurrency of another economy. For example, if a US auto importer wants to buy aGerman car, the importer must buy euros. The law of demand holds: as the price ofa foreign currency increases, the quantity of that currency demanded will decrease.

Foreign currencies are supplied by foreign households, firms, and governments thatwish to purchase goods, services, or financial assets denominated in the domesticcurrency. For example, if a Canadian bank wants to buy a US government bond, thebank must sell Canadian dollars. As the price of a foreign currency increases, thequantity supplied of that currency increases.

Exchange rates are determined just like other prices—by the interaction of supplyand demand. At the equilibrium exchange rate, the supply and demand for acurrency are equal. Shifts in the supply or the demand for a currency lead tochanges in the exchange rate. Because one currency is exchanged for another in aforeign exchange market, the demand for one currency entails the supply ofanother. Thus the dollar market for euros (where the price is dollars per euro andthe quantity is euros) is the mirror image of the euro market for dollars (where theprice is euros per dollar and the quantity is dollars).

To be concrete, consider the demand for and the supply of euros. The supply ofeuros comes from the following:

• European households and firms that wish to buy goods and servicesfrom countries that do not have the euro as their currency

• European investors who wish to buy assets (government debt, stocks,bonds, etc.) that are denominated in currencies other than the euro

The demand for euros comes from the following:


799

• Households and firms in noneuro countries that wish to buy goods andservices from Europe

• Investors in noneuro countries that wish to buy assets (governmentdebt, stocks, bonds, etc.) that are denominated in euros

Figure 16.7 "The Foreign Exchange Market" shows the dollar market for euros. Onthe horizontal axis is the quantity of euros traded. On the vertical axis is the pricein terms of dollars. The intersection of the supply and demand curves determinesthe equilibrium exchange rate.

Figure 16.7 The Foreign Exchange Market

The foreign exchange market can be used as a basis for comparative staticsexercises. We can study how changes in an economy affect the exchange rate. Forexample, suppose there is an increase in the level of economic activity in the UnitedStates. This leads to an increase in the demand for European goods and services. Tomake these purchases, US households and firms will demand more euros. Thiscauses an outward shift in the demand curve and an increase in the dollar price ofeuros.


16.10 Foreign Exchange Market 800

When the dollar price of a euro increases, we say that the dollar has depreciatedrelative to the euro. From the perspective of the euro, the depreciation of the dollarrepresents an appreciation of the euro.

Key Insight

• As the exchange rate increases (so a currency becomes more valuable),a greater quantity of the currency is supplied to the market and asmaller quantity is demanded.


• Chapter 4 "The Interconnected Economy"• Chapter 7 "The Great Depression"• Chapter 9 "Money: A User’s Guide"• Chapter 10 "Understanding the Fed"


16.10 Foreign Exchange Market 801

16.11 Growth Rates

If some variable x (for example, the number of gallons of gasoline sold in a week)changes from x1 to x2, then we can define the change in that variable as Δx = x2 − x1.

But there are difficulties with this simple definition. The number that we calculatewill change, depending on the units in which we measure x. If we measure inmillions of gallons, x will be a much smaller number than if we measure in gallons.If we measured x in liters rather than gallons (as it is measured in most countries),it would be a bigger number. So the number we calculate depends on the units wechoose. To avoid these problems, we look at percentage changes and express thechange as a fraction of the individual value. In what follows, we use the notation%Δx to mean the percentage change in x and define it as follows: %Δx = (x2 − x1)/x1.

A percentage change equal to 0.1 means that gasoline consumption increased by 10percent. Why? Because 10 percent means 10 “per hundred,” so 10 percent = 10/100= 0.1.

Very often in economics, we are interested in changes that take place over time.Thus we might want to compare gross domestic product (GDP) between 2012 and2013. Suppose we know that GDP in the United States in 2012 was $14 trillion andthat GDP in 2013 was $14.7 trillion. Using the letter Y to denote GDP measured intrillions, we write Y2012 = 14.0 and Y2013 = 14.7. If we want to talk about GDP at

different points in time without specifying a particular year, we use the notation Yt.

We express the change in a variable over time in the form of a growth rate, which isjust an example of a percentage change. Thus the growth rate of GDP in 2013 iscalculated as follows:

%ΔY2013 = (Y2013 − Y2012)/Y2012 = (14.7 − 14)/14 = 0.05.

The growth rate equals 5 percent. In general, we write %ΔYt+1 = (Yt+1 − Yt)/Yt.

Occasionally, we use the gross growth rate, which simply equals 1 + the growth rate.So, for example, the gross growth rate of GDP equals Y2013/Y2012, or 1.05.

There are some useful rules that describe the behavior of percentage changes andgrowth rates.

The Product Rule. Suppose we have three variables, x, y, and z, and suppose

x = yz.


802

Then

%Δx = %Δy + %Δz.

In other words, the growth rate of a product of two variables equals the sum of thegrowth rates of the individual variables.

The Quotient Rule. Now suppose we rearrange our original equation by dividingboth sides by z to obtain

If we take the product rule and subtract %Δz from both sides, we get the following:

%Δy = %Δx − %Δz.

The Power Rule. There is one more rule of growth rates that we make use of insome advanced topics, such as growth accounting. Suppose that

y = xa.

Then

%Δy = a(%Δx).

For example, if y = x2, then the growth rate of y is twice the growth rate of x. Ify = x

⎯⎯√ , then the growth rate of y is half the growth rate of x (remembering that a

square root is the same as a power of ½).

More Formally

Growth rates compound over time: if the growth rate of a variable is constant, thenthe change in the variable increases over time. For example, suppose GDP in 2020 is20.0, and it grows at 10 percent per year. Then in 2021, GDP is 22.0 (an increase of2.0), but in 2022, GDP is 24.2 (an increase of 2.2). If this compounding takes placeevery instant, then we say that we have exponential growth. Formally, we writeexponential growth using the number e = 2.71828.… If the value of Y at time 0 equalsY0 and if Y grows at the constant rate g (where g is an “annualized” or per year

growth rate), then at time t (measured in years),

y =x

z.


16.11 Growth Rates 803

Yt = egtY0.

A version of this formula can also be used to calculate the average growth rate of avariable if we know its value at two different times. We can write the formula as

egt = Yt/Y0,

which also means

gt = ln(Yt/Y0),

where ln() is the natural logarithm. You do not need to know exactly what thismeans; you can simply calculate a logarithm using a scientific calculator or aspreadsheet. Dividing by t we get the average growth rate

g = ln(Yt/Y0)/t.

For example, suppose GDP in 2020 is 20.0 and GDP in 2030 is 28.0. Then Y2030/Y2020 =

28/20 = 1.4. Using a calculator, we can find ln(1.4) = 0.3364. Dividing by 10 (since thetwo dates are 10 years apart), we get an average growth rate of 0.034, or 3.4 percentper year.


• Chapter 3 "The State of the Economy"• Chapter 5 "Globalization and Competitiveness"• Chapter 6 "Global Prosperity and Global Poverty"• Chapter 7 "The Great Depression"• Chapter 9 "Money: A User’s Guide"• Chapter 11 "Inflations Big and Small"


16.11 Growth Rates 804

16.12 Mean and Variance

To start our presentation of descriptive statistics, we construct a data set using aspreadsheet program. The idea is to simulate the flipping of a two-sided coin.Although you might think it would be easier just to flip a coin, doing this on aspreadsheet gives you a full range of tools embedded in that program. To generatethe data set, we drew 10 random numbers using the spreadsheet program. In theprogram we used, the function was called RAND and this generated the choice of anumber between zero and one. Those choices are listed in the second column ofTable 16.9.

The third column creates the two events of heads and tails that we normallyassociate with a coin flip. To generate this last column, we adopted a rule: if therandom number was less than 0.5, we termed this a “tail” and assigned a 0 to thedraw; otherwise we termed it a “head” and assigned a 1 to the draw. The choice of0.5 as the cutoff for heads reflects the fact that we are considering the flips of a faircoin in which each side has the same probability: 0.5.

Table 16.9

Draw Random Number Heads (1) or Tails (0)

1 0.94 1

2 0.84 1

3 0.26 0

4 0.04 0

5 0.01 0

6 0.57 1

7 0.74 1

8 0.81 1

9 0.64 1

10 0.25 0

Keep in mind that the realization of the random number in draw i is independent ofthe realizations of the random numbers in both past and future draws. Whether a


805

coin comes up heads or tails on any particular flip does not depend on otheroutcomes.

There are many ways to summarize the information contained in a sample of data.Even before you start to compute some complicated statistics, having a way topresent the data is important. One possibility is a bar graph in which the fraction ofobservations of each outcome is easily shown. Alternatively, a pie chart is oftenused to display this fraction. Both the pie chart and the bar diagram are commonlyfound in spreadsheet programs.

Economists and statisticians often want to describe data in terms of numbers ratherthan figures. We use the data from the table to define and illustrate two statisticsthat are commonly used in economics discussions. The first is the mean (or average)and is a measure of central tendency. Before you read any further, ask, “What do youthink the average ought to be from the coin flipping exercise?” It is natural to say0.5, since half the time the outcome will be a head and thus have a value of zero,whereas the remainder of the time the outcome will be a tail and thus have a valueof one.

Whether or not that guess holds can be checked by looking at Table 16.9 andcalculating the mean of the outcome. We let ki be the outcome of draw i. For

example, from the table, k1 = 1 and k5 = 0. Then the formula for the mean if there

are N draws is μ = Σiki/N. Here Σiki means the sum of the ki outcomes. In words, the

mean, denoted by μ, is calculated by adding together the draws and dividing by thenumber of draws (N). In the table, N = 10, and the sum of the draws of randomnumbers is about 51.0. Thus the mean of the 10 draws is about 0.51.

We can also calculate the mean of the heads/tails column, which is 0.6 since headscame up 6 times in our experiment. This calculation of the mean differs from themean of the draws since the numbers in the two columns differ with the thirdcolumn being a very discrete way to represent the information in the secondcolumn.

A second commonly used statistic is a measure of dispersion of the data called thevariance. The variance, denoted σ2, is calculated as σ2 = Σi(ki − μ)2/(N). From this

formula, if all the draws were the same (thus equal to the mean), then the variancewould be zero. As the draws spread out from the mean (both above and below), thevariance increases. Since some observations are above the mean and others below,we square the difference between a single observation (ki) and the mean (μ) when

calculating the variance. This means that values above and below the mean both


16.12 Mean and Variance 806

contribute a positive amount to the variance. Squaring also means that values thatare a long way away from the mean have a big effect on the variance.

For the data given in the table, the mean of the 10 draws was given as μ = 0.51. So tocalculate the variance, we would subtract the mean from each draw, square thedifference, and then add together the squared differences. This yields a variance of0.118 for this draw. A closely related concept is that of the standard deviation,which is the square root of the variance. For our example, the standard deviation is0.34. The standard deviation is greater than the variance since the variance is lessthan 1.


• Chapter 5 "Globalization and Competitiveness"• Chapter 7 "The Great Depression"• Chapter 11 "Inflations Big and Small"


16.12 Mean and Variance 807

16.13 Correlation and Causality

Correlation is a statistical measure describing how two variables move together. Incontrast, causality (or causation) goes deeper into the relationship between twovariables by looking for cause and effect.

Correlation is a statistical property that summarizes the way in which two variablesmove either over time or across people (firms, governments, etc.). The concept ofcorrelation is quite natural to us, as we often take note of how two variablesinterrelate. If you think back to high school, you probably have a sense of how yourclassmates did in terms of two measures of performance: grade point average (GPA)and the results on a standardized college entrance exam (SAT or ACT). It is likelythat classmates with high GPAs also had high scores on the SAT or ACT exam. In thisinstance, we would say that the GPA and SAT/ACT scores were positively correlated:looking across your classmates, when a person’s GPA is higher than average, thatperson’s SAT or ACT score is likely to be higher than average as well.

As another example, consider the relationship between a household’s income andits expenditures on housing. If you conducted a survey across households, it islikely that you would find that richer households spend more on most goods andservices, including housing. In this case, we would conclude that income andexpenditures on housing are positively correlated.

When economists look at data for a whole economy, they often focus on a measureof how much is produced, which we call real gross domestic product (real GDP), andthe fraction of workers without jobs, called the unemployment rate. Over longperiods of time, when GDP is above average (the economy is doing well), theunemployment rate is below average. In this case, GDP and the unemployment rateare negatively correlated, as they tend to move in opposite directions.

The fact that one variable is correlated with another does not inform us aboutwhether one variable causes the other. Imagine yourself on an airplane in a relaxedmood, reading or listening to music. Suddenly, the pilot comes on the publicaddress system and requests that you buckle your seat belts. Usually, such a requestis followed by turbulence. This is a correlation: the announcement by the pilot ispositively correlated with air turbulence. The correlation is of course not perfectbecause sometimes you hit some bumps without warning, and sometimes the pilot’sannouncement is not followed by turbulence.


808

But—obviously—this does not mean that we could solve the turbulence problem byturning off the public address system. The pilot’s announcement does not cause theturbulence. The turbulence is there whether the pilot announces it or not. In fact,the causality runs the other way. The turbulence causes the pilot’s announcement.

We noted earlier that real GDP and unemployment are negatively correlated. Whenreal GDP is below average, as it is during a recession, the unemployment rate istypically above average. But what is the causality here? If unemployment causedrecessions, we might be tempted to adopt a policy that makes unemploymentillegal. For example, the government could fine firms if they lay off workers. This isnot a good policy because we do not think that low unemployment causes high realGDP. Neither do we necessarily think that high real GDP causes low unemployment.Instead, based on economic theory, there are other influences that affect both realGDP and unemployment.

More Formally

Suppose you have N observations of two variables, x and y, where xi and yi are the

values of these variables in observation i = 1, 2,…, N. The mean of x, denoted μx, is

the sum over the values of x in the sample is divided by N; the scenario applies for y.

and

We can also calculate the variance and standard deviations of x and y. Thecalculation for the variance of x, denoted σ 2

x , is as follows:

The standard deviation of x is the square root of σ 2x :

μx =x1 + x2 + ... + xN

N

μy =y1 + y2 + ... + yN

N.

σ 2x =

(x1 − μx )2 + (x2 − μx )2 + ... (xN − μx )2

N.

σx =(x1 − μx )2 + (x2 − μx )2 + ... (xN − μx )2

N

⎯ ⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯

√ .


16.13 Correlation and Causality 809

With these ingredients, the correlation of (x,y), denoted corr(x,y), is given by


• Chapter 7 "The Great Depression"• Chapter 11 "Inflations Big and Small"• Chapter 14 "Balancing the Budget"

corr(x, y) =(x1 − μx )(y1 − μy ) + (x2 − μx )(y2 − μy ) + ... (xN − μx )(yN − μy )

Nσxσy.


16.13 Correlation and Causality 810

16.14 The Fisher Equation: Nominal and Real Interest Rates

When you borrow or lend, you normally do so in dollar terms. If you take out a loan,the loan is denominated in dollars, and your promised payments are denominatedin dollars. These dollar flows must be corrected for inflation to calculate therepayment in real terms. A similar point holds if you are a lender: you need tocalculate the interest you earn on saving by correcting for inflation.

The Fisher equation provides the link between nominal and real interest rates. Toconvert from nominal interest rates to real interest rates, we use the followingformula:

real interest rate ≈ nominal interest rate − inflation rate.

To find the real interest rate, we take the nominal interest rate and subtract theinflation rate. For example, if a loan has a 12 percent interest rate and the inflationrate is 8 percent, then the real return on that loan is 4 percent.

In calculating the real interest rate, we used the actual inflation rate. This isappropriate when you wish to understand the real interest rate actually paid undera loan contract. But at the time a loan agreement is made, the inflation rate thatwill occur in the future is not known with certainty. Instead, the borrower andlender use their expectations of future inflation to determine the interest rate on aloan. From that perspective, we use the following formula:

contracted nominal interest rate ≈ real interest rate + expected inflation rate.

We use the term contracted nominal interest rate to make clear that this is the rate setat the time of a loan agreement, not the realized real interest rate.

Key Insight

• To correct a nominal interest rate for inflation, subtract the inflationrate from the nominal interest rate.

More Formally

Imagine two individuals write a loan contract to borrow P dollars at a nominalinterest rate of i. This means that next year the amount to be repaid will be P × (1 +i). This is a standard loan contract with a nominal interest rate of i.


811

Now imagine that the individuals decided to write a loan contract to guarantee aconstant real return (in terms of goods not dollars) denoted r. So the contractprovides P this year in return for being repaid (enough dollars to buy) (1 + r) units ofreal gross domestic product (real GDP) next year. To repay this loan, the borrowergives the lender enough money to buy (1 + r) units of real GDP for each unit of realGDP that is lent. So if the inflation rate is π, then the price level has risen to P × (1 +π), so the repayment in dollars for a loan of P dollars would be P(1 + r) × (1 + π).

Here (1 + π) is one plus the inflation rate. The inflation rate πt+1 is defined—as

usual—as the percentage change in the price level from period t to period t + 1.

πt+1 = (Pt+1 − Pt)/Pt.

If a period is one year, then the price level next year is equal to the price this yearmultiplied by (1 + π):

Pt+1 = (1 + πt) × Pt.

The Fisher equation says that these two contracts should be equivalent:

(1 + i) = (1 + r) × (1 + π).

As an approximation, this equation implies

i ≈ r + π.

To see this, multiply out the right-hand side and subtract 1 from each side to obtain

i = r + π + rπ.

If r and π are small numbers, then rπ is a very small number and can safely beignored. For example, if r = 0.02 and π = 0.03, then rπ = 0.0006, and ourapproximation is about 99 percent accurate.


• Chapter 4 "The Interconnected Economy"• Chapter 9 "Money: A User’s Guide"• Chapter 10 "Understanding the Fed"• Chapter 11 "Inflations Big and Small"


16.14 The Fisher Equation: Nominal and Real Interest Rates 812

16.15 The Aggregate Production Function

The aggregate production function describes how total real gross domestic product(real GDP) in an economy depends on available inputs. Aggregate output (real GDP)depends on the following:

• Physical capital—machines, production facilities, and so forth that areused in production

• Labor—the number of hours that are worked in the entire economy• Human capital—skills and education embodied in the workforce of the

economy• Knowledge—basic scientific knowledge, and blueprints that describe

the available production processes• Social infrastructure—the general business, legal and cultural

environment• The amount of natural resources available in an economy• Anything else that we have not yet included

We group the inputs other than labor, physical, and human capital together, andcall them technology.

The aggregate production function has several key properties. First, outputincreases when there are increases in physical capital, labor, and natural resources.In other words, the marginal products of these inputs are all positive.

Second, the increase in output from adding more inputs is lower when we havemore of a factor. This is called diminishing marginal product. That is,

• The more capital we have, the less additional output we obtain fromadditional capital.

• The more labor we have, the less additional output we obtain fromadditional labor.

• The more natural resources we have, the less additional output weobtain from additional resources.

In addition, increases in output can also come from increases in human capital,knowledge, and social infrastructure. In contrast to capital and labor, we do notassume that there are diminishing returns to human capital and technology. Onereason is that we do not have a natural or an obvious measure for human capital,


813

knowledge, or social infrastructure, whereas we do for labor and capital (hours ofwork and hours of capital usage).

Figure 16.8 shows the relationship between output and capital, holding fixed thelevel of other inputs. This figure shows two properties of the aggregate productionfunction. As capital input is increased, output increases as well. But the change inoutput obtained by increasing the capital stock is lower when the capital stock ishigher: this is the diminishing marginal product of capital.

Figure 16.8

In many applications, we want to understand how the aggregate productionfunction responds to variations in the technology or other inputs. This is illustratedin Figure 16.9. An increase in, say, technology means that for a given level of thecapital stock, more output is produced: the production function shifts upward astechnology increases. Further, as technology increases, the production function issteeper: the increase in technology increases the marginal product of capital.


16.15 The Aggregate Production Function 814

Figure 16.9

Key Insight

• The aggregate production function allows us to determine the outputof an economy given inputs of capital, labor, human capital, andtechnology.

More FormallySpecific Forms for the Production Function

We can write the production function in mathematical form. We use Y to representreal GDP, K to represent the physical capital stock, L to represent labor, H torepresent human capital, and A to represent technology (including naturalresources). If we want to speak about production completely generally, then we canwrite Y = F(K,L,H,A). Here F() means “some function of.”

A lot of the time, economists work with a production function that has a specificmathematical form, yet is still reasonably simple:

Y = A × Ka × (L × H)(1 − a),



where a is just a number. This is called a Cobb-Douglas production function. Itturns out that this production function does a remarkably good job of summarizingaggregate production in the economy. In fact, we also know that we can describeproduction in actual economies quite well if we suppose that a = 1/3.


• Chapter 5 "Globalization and Competitiveness"• Chapter 6 "Global Prosperity and Global Poverty"• Chapter 7 "The Great Depression"



16.16 The Circular Flow of Income

The circular flow of income describes the flows of money among the five mainsectors of an economy. As individuals and firms buy and sell goods and services,money flows among the different sectors of an economy. The circular flow ofincome describes these flows of dollars (pesos, euros, or whatever). From a simpleversion of the circular flow, we learn that—as a matter of accounting—

gross domestic product (GDP) = income = production = spending.

This relationship lies at the heart of macroeconomic analysis.

There are two sides to every transaction. Corresponding to the flows of money inthe circular flow, there are flows of goods and services among these sectors. Forexample, the wage income received by consumers is in return for labor servicesthat flow from households to firms. The consumption spending of households is inreturn for the goods and services that flow from firms to households.

A complete version of the circular flow is presented in Figure 16.10. (Chapter 3 "TheState of the Economy" contains a discussion of a simpler version of the circular flowwith only two sectors: households and firms.)


817

Figure 16.10

The complete circular flow has five sectors: a household sector, a firm sector, agovernment sector, a foreign sector, and a financial sector. Different chapters of thebook emphasize different pieces of the circular flow, and Figure 16.10 shows us howeverything fits together. In the following subsections, we look at the flows into andfrom each sector in turn. In each case, the balance of the flows into and from eachsector underlies a useful economic relationship.

The Firm Sector

Figure 16.10 includes the component of the circular flow associated with the flowsinto and from the firm sector of an economy. We know that the total flow of dollarsfrom the firm sector measures the total value of production in an economy. Thetotal flow of dollars into the firm sector equals total expenditures on GDP. Wetherefore know that

production = consumption + investment + government purchases + net exports.

This equation is called the national income identity and is the most fundamentalrelationship in the national accounts.


16.16 The Circular Flow of Income 818

By consumption we mean total consumption expenditures by households on finalgoods and services. Investment refers to the purchase of goods and services that, inone way or another, help to produce more output in the future. Governmentpurchases include all purchases of goods and services by the government. Netexports, which equal exports minus imports, measure the expenditure flowsassociated with the rest of the world.

The Household Sector

The household sector summarizes the behavior of private individuals in their rolesas consumers/savers and suppliers of labor. The balance of flows into and from thissector is the basis of the household budget constraint. Households receive incomefrom firms, in the form of wages and in the form of dividends resulting from theirownership of firms. The income that households have available to them after alltaxes have been paid to the government and all transfers received is calleddisposable income. Households spend some of their disposable income and save therest. In other words,

disposable income = consumption + household savings.

This is the household budget constraint. In Figure 16.10, this equationcorresponds to the fact that the flows into and from the household sector mustbalance.

The Government Sector

The government sector summarizes the actions of all levels of government in aneconomy. Governments tax their citizens, pay transfers to them, and purchasegoods from the firm sector of the economy. Governments also borrow from or lendto the financial sector. The amount that the government collects in taxes need notequal the amount that it pays out for government purchases and transfers. If thegovernment spends more than it gathers in taxes, then it must borrow from thefinancial markets to make up the shortfall.

The circular flow figure shows two flows into the government sector and two flowsout. Since the flows into and from the government sector must balance, we knowthat

government purchases + transfers = tax revenues + government borrowing.

Government borrowing is sometimes referred to as the government budget deficit.This equation is the government budget constraint.



Some of the flows in the circular flow can go in either direction. When thegovernment is running a deficit, there is a flow of dollars to the government sectorfrom the financial markets. Alternatively, the government may run a surplus,meaning that its revenues from taxation are greater than its spending on purchasesand transfers. In this case, the government is saving rather than borrowing, andthere is a flow of dollars to the financial markets from the government sector.

The Foreign Sector

The circular flow includes a country’s dealings with the rest of the world. Theseflows include exports, imports, and borrowing from other countries. Exports aregoods and services produced in one country and purchased by households, firms,and governments of another country. Imports are goods and services purchased byhouseholds, firms, and governments in one country but produced in anothercountry. Net exports are exports minus imports. When net exports are positive, acountry is running a trade surplus: exports exceed imports. When net exports arenegative, a country is running a trade deficit: imports exceed exports. The thirdflow between countries is borrowing and lending. Governments, individuals, andfirms in one country may borrow from or lend to another country.

Net exports and borrowing are linked. If a country runs a trade deficit, it borrowsfrom other countries to finance that deficit. If we look at the flows into and fromthe foreign sector, we see that

borrowing from other countries + exports = imports.

Subtracting exports from both sides, we obtain

borrowing from other countries = imports − exports = trade deficit.

Whenever our economy runs a trade deficit, we are borrowing from othercountries. If our economy runs a trade surplus, then we are lending to othercountries.

This analysis has omitted one detail. When we lend to other countries, we acquiretheir assets, so each year we get income from those assets. When we borrow fromother countries, they acquire our assets, so we pay them income on those assets.Those income flows are added to the trade surplus/deficit to give the current accountof the economy. It is the current account that must be matched by borrowing fromor lending to other countries. A positive current account means that net exportsplus net income flows from the rest of the world are positive. In this case, oureconomy is lending to the rest of the world and acquiring more assets.



The Financial Sector

The financial sector of an economy summarizes the behavior of banks and otherfinancial institutions. The balance of flows into and from the financial sector tell usthat investment is financed by national savings and borrowing from abroad. Thefinancial sector is at the heart of the circular flow. The figure shows four flows intoand from the financial sector.

1. Households divide their after-tax income between consumption andsavings. Thus any income that they receive today but wish to put asidefor the future is sent to the financial markets. The household sector asa whole saves so, on net, there is a flow of dollars from the householdsector into the financial markets.

2. The flow of money from the financial sector into the firm sectorprovides the funds that are available to firms for investment purposes.

3. The flow of dollars between the financial sector and the governmentsector reflects the borrowing (or lending) of governments. The flowcan go in either direction. When government expenditures exceedgovernment revenues, the government must borrow from the privatesector, and there is a flow of dollars from the financial sector to thegovernment. This is the case of a government deficit. When thegovernment’s revenues are greater than its expenditures, by contrast,there is a government surplus and a flow of dollars into the financialsector.

4. The flow of dollars between the financial sector and the foreign sectorcan also go in either direction. An economy with positive net exports islending to other countries: there is a flow of money from an economy.An economy with negative net exports (a trade deficit) is borrowingfrom other countries.

The national savings of the economy is the savings carried out by the private andgovernment sectors taken together. When the government is running a deficit,some of the savings of households and firms must be used to fund that deficit, sothere is less left over to finance investment. National savings is then equal toprivate savings minus the government deficit—that is, private savings minusgovernment borrowing:

national savings = private savings − government borrowing.

If the government is running a surplus, then

national savings = private savings + government surplus.



National savings is therefore the amount that an economy as a whole saves. It isequal to what is left over after we subtract consumption and government spendingfrom GDP. To see this, notice that

private savings − government borrowing = income − taxes + transfers − consumption− (government purchases + transfers − taxes)

= income − consumption − government purchases.

This is the domestic money that is available for investment.

If we are borrowing from other countries, there is another source of funds forinvestment. The flows into and from the financial sector must balance, so

investment = national savings + borrowing from other countries.

Conversely, if we are lending to other countries, then our national savings isdivided between investment and lending to other countries:

national savings = investment + lending to other countries.


• Chapter 3 "The State of the Economy"• Chapter 4 "The Interconnected Economy"• Chapter 6 "Global Prosperity and Global Poverty"• Chapter 7 "The Great Depression"• Chapter 9 "Money: A User’s Guide"• Chapter 11 "Inflations Big and Small"• Chapter 12 "Income Taxes"• Chapter 14 "Balancing the Budget"• Chapter 15 "The Global Financial Crisis"



16.17 Growth Accounting

Growth accounting is a tool that tells us how changes in real gross domestic product(real GDP) in an economy are due to changes in available capital, labor, humancapital, and technology. Economists have shown that, under reasonably generalcircumstances, the change in output in an economy can be written as follows:

output growth rate = a × capital stock growth rate + [(1 − a) × labor hours growthrate]+ [(1 − a) × human capital growth rate] + technology growth rate.

In this equation, a is just a number. For example, if a = 1/3, the growth in output isas follows:

output growth rate = (1/3 × capital stock growth rate) + (2/3 × labor hours growthrate)+ (2/3 × human capital growth rate) + technology growth rate.

Growth rates can be positive or negative, so we can use this equation to analyzedecreases in GDP as well as increases. This expression for the growth rate of output,by the way, is obtained by applying the rules of growth rates (discussed in Section16.11 "Growth Rates") to the Cobb-Douglas aggregate production function(discussed in Section 16.15 "The Aggregate Production Function").

What can we measure in this expression? We can measure the growth in output, thegrowth in the capital stock, and the growth in labor hours. Human capital is moredifficult to measure, but we can use information on schooling, literacy rates, and soforth. We cannot, however, measure the growth rate of technology. So we use thegrowth accounting equation to infer the growth in technology from the things wecan measure. Rearranging the growth accounting equation,

technology growth rate = output growth rate − (a × capital stock growth rate)− [(1 −a) × labor hours growth rate] − [(1 − a) × human capital growth rate].

So if we know the number a, we are done—we can use measures of the growth inoutput, labor, capital stock, and human capital to solve for the technology growthrate. In fact, we do have a way of measuring a. The technical details are notimportant here, but a good measure of (1 − a) is simply the total payments to laborin the economy (that is, the total of wages and other compensation) as a fraction ofoverall GDP. For most economies, a is in the range of about 1/3 to 1/2.


823

Key Insight

• The growth accounting tool allows us to determine the contributionsof the various factors of economic growth.


• Chapter 5 "Globalization and Competitiveness"• Chapter 6 "Global Prosperity and Global Poverty"• Chapter 7 "The Great Depression"


16.17 Growth Accounting 824

16.18 The Solow Growth Model

The analysis in Chapter 6 "Global Prosperity and Global Poverty" is (implicitly)based on a theory of economic growth known as the Solow growth model. Here wepresent two formal versions of the mathematics of the model. The first takes as itsfocus the capital accumulation equation and explains how the capital stock evolvesin the economy. This version ignores the role of human capital and ignores thelong-run growth path of the economy. The second follows the exposition of thechapter and is based around the derivation of the balanced growth path. They are,however, simply two different ways of approaching the same problem.

Presentation 1

There are three components of this presentation of the model: technology, capitalaccumulation, and saving. The first component of the Solow growth model is thespecification of technology and comes from the aggregate production function. Weexpress output per worker (y) as a function of capital per worker (k) and technology(A). A mathematical expression of this relationship is

y = Af(k),

where f(k) means that output per worker depends on capital per worker. As in ourpresentation of production functions, output increases with technology. We assumethat f() has the properties that more capital leads to more output per capita at adiminishing rate. As an example, suppose

y = Ak1/3.

In this case the marginal product of capital is positive but diminishing.

The second component is capital accumulation. If we let kt be the amount of capital

per capita at the start of year t, then we know that

kt+1 = kt(1 − δ) + it.

This expression shows how the capital stock changes over time. Here δ is the rate ofphysical depreciation so that between year t and year t +1, δkt units of capital are

lost from depreciation. But during year t, there is investment (it) that yields new

capital in the following year.


825

The final component of the Solow growth model is saving. In a closed economy,saving is the same as investment. Thus we link it in the accumulation equation to

saving. Assume that saving per capita (st) is given by

st = s × yt.

Here s is a constant between zero and one, so only a fraction of total output is saved.

Using the fact that savings equals investment, along with the per capita productionfunction, we can relate investment to the level of capital:

it = sAf(kt).

We can then write the equation for the evolution of the capital stock as follows:

kt+1 = kt(1 − δ) + sAf(kt).

Once we have specified the function f(), we can follow the evolution of the capitalstock over time. Generally, the path of the capital stock over time has twoimportant properties:

1. Steady state. There is a particular level of the capital stock such that ifthe economy accumulates that amount of capital, it stays at that levelof capital. We call this the steady state level of capital, denoted k*.

2. Stability. The economy will tend toward the per capita capital stockk*.

To be more specific, the steady state level of capital solves the following equation:

k* = k*(1 − δ) + sAf(k*).

At the steady state, the amount of capital lost by depreciation is exactly offset bysaving. This means that at the steady state, net investment is exactly zero. Theproperty of stability means that if the current capital stock is below k*, theeconomy will accumulate capital so that kt+1 > kt. And if the current capital stock is

above k*, the economy will decumulate capital so that kt+1 < kt.

If two countries share the same technology (A) and the same production function[f(k)], then over time these two countries will eventually have the same stock ofcapital per worker. If there are differences in the technology or the production


16.18 The Solow Growth Model 826

function, then there is no reason for the two countries to converge to the same levelof capital stock per worker.

Presentation 2

In this presentation, we explain the balanced-growth path of the economy andprove some of the claims made in the text. The model takes as given (exogenous)the investment rate; the depreciation rate; and the growth rates of the workforce,human capital, and technology. The endogenous variables are output and physicalcapital stock.

The notation for the presentation is given in Table 16.10 "Notation in the SolowGrowth Model": We use the notation gx to represent the growth rate of a variable x;

that is, gx = Δxx = %Δx.

There are two key ingredients to the model: the aggregate production function andthe equation for capital accumulation.

Table 16.10 Notation in the Solow Growth Model

Variable Symbol

Real gross domestic product Y

Capital stock K

Human capital H

Workforce L

Technology A

Investment rate i

Depreciation rate δ

The Production Function

The production function we use is the Cobb-Douglas production function:

Equation 16.1

Y = Ka(HL)1−aA.



Growth Accounting

If we apply the rules of growth rates to Equation 16.1, we get the followingexpression:

Equation 16.2

gY = agK + (1 − a)(gL + gH) + gA.Balanced Growth

The condition for balanced growth is that gY = gK. When we impose this condition

on our equation for the growth rate of output (Equation 16.2), we get

where the superscript “BG” indicates that we are considering the values of variableswhen the economy is on a balanced growth path. This equation simplifies to

Equation 16.3

The growth in output on a balanced-growth path depends on the growth rates ofthe workforce, human capital, and technology.

Using this, we can rewrite Equation 16.2 as follows:

Equation 16.4

The actual growth rate in output is an average of the balanced-growth rate ofoutput and the growth rate of the capital stock.

Capital Accumulation

The second piece of our model is the capital accumulation equation. The growthrate of the capital stock is given by

gBGY = agBGY + (1 − a)(gL + gH) + gA ,

gBGY = gL + gH + ( 11 − a) gA .

gY = agK + (1 − a)gBGY .



Equation 16.5

Divide the numerator and denominator of the first term by Y, remembering that i =I/Y.

Equation 16.6

The growth rate of the capital stock depends positively on the investment rate andnegatively on the depreciation rate. It also depends negatively on the currentcapital-output ratio.

The Balanced-Growth Capital-Output Ratio

Now rearrange Equation 16.6 to give the ratio of capital to gross domestic product(GDP), given the depreciation rate, the investment rate, and the growth rate of thecapital stock:

When the economy is on a balanced growth path, gK = gBGY , so

We can also substitute in our balanced-growth expression for gBGY (Equation 16.3) toget an expression for the balanced-growth capital output ratio in terms ofexogenous variables.

gK =I

K− δ.

gK =i

K/Y− δ.

K

Y=

i

δ + gK.

(K

Y )BG

=i

δ + gBGY.

(K

Y )BG

=i

δ + gL + gH + 11−a gA

.



Convergence

The proof that economies will converge to the balanced-growth ratio of capital to

GDP is relatively straightforward. We want to show that if K/Y < ( KY )BG, then

capital grows faster than output. If capital is growing faster than output, gK − gY > 0.

First, go back to Equation 16.4:

Subtract both sides from the growth rate of capital:

Now compare the general expression for ratio of capital to GDP with its balancedgrowth value:

and

If K/Y < ( KY )BG, then it must be the case that gK > gBGY , which implies (from the

previous equation) that gK > gY.

Output per Worker Growth

If we want to examine the growth in output per worker rather than total output, wetake the per-worker production function (Equation 16.2) and apply the rules ofgrowth rates to that equation.

(1 − a)gY = a[gK − gY] + (1 − a)[gL + gH] + gA = a[gK − gY] + (1 − a)[gL + gH] + gA.

We then we divide by (1 − a) to get

gY = agK + (1 − a)gBGY .

gK − gY = gK − agK − (1 − a)gBGY = (1 − a) (gK − gBGY ) .

K/Y =i

δ + gK(general expression)

(K/Y) BG =i

δ + gBGY(balanced growth).

gY/L = ( a

1 − a) gK/Y + gH + gA .



and subtract gL from each side to obtain

Finally, we note that gY − gL = gY/L:

With balanced growth, the first term is equal to zero, so

Endogenous Investment Rate

In this analysis, we made the assumption from the Solow model that the investmentrate is constant. The essential arguments that we have made still apply if theinvestment rate is higher when the marginal product of capital is higher. Theargument for convergence becomes stronger because a low value of K/Y implies ahigher marginal product of capital and thus a higher investment rate. Thisincreases the growth rate of capital and causes an economy to converge morequickly to its balanced-growth path.

Endogenous Growth

Take the production function

Y = Ka(HL)1−aA.

Now assume A is constant and H = ( BA )1/(1−a) × (K/L), so

gY =a

(1 − a)[gK − gY ] + gL + gH + ( 1

1 − a) gA

gY − gL =a

(1 − a)[gK − gY ] + gH + ( 1

1 − a) gA .

gY/L =a

(1 − a)[gK − gY ] + gH + ( 1

1 − a) gA .

gBGY/L = gH + ( 11 − a) gA .




• Chapter 5 "Globalization and Competitiveness"• Chapter 6 "Global Prosperity and Global Poverty"

Y =

=

=

K a(L( B

A )1/(1−a)

(K/L))1−a

A

K a(( B

A )1/(1−a)

(K))1−a

A

BK.



16.19 The Aggregate Expenditure Model

The aggregate expenditure model relates the components of spending(consumption, investment, government purchases, and net exports) to the level ofeconomic activity. In the short run, taking the price level as fixed, the level ofspending predicted by the aggregate expenditure model determines the level ofeconomic activity in an economy.

An insight from the circular flow is that real gross domestic product (real GDP)measures three things: the production of firms, the income earned by households,and total spending on firms’ output. The aggregate expenditure model focuses onthe relationships between production (GDP) and planned spending:

GDP = planned spending

= consumption + investment + government purchases + net exports.

Planned spending depends on the level of income/production in an economy, forthe following reasons:

• If households have higher income, they will increase their spending.(This is captured by the consumption function.)

• Firms are likely to decide that higher levels of production—particularlyif they are expected to persist—mean that they should build up theircapital stock and should thus increase their investment.

• Higher income means that domestic consumers are likely to spendmore on imported goods. Since net exports equal exports minusimports, higher imports means lower net exports.

The negative net export link is not large enough to overcome the other positivelinks, so we conclude that when income increases, so also does plannedexpenditure. We illustrate this in Figure 16.11 "Planned Spending in the AggregateExpenditure Model" where we suppose for simplicity that there is a linearrelationship between spending and GDP. The equation of the line is as follows:

spending = autonomous spending + marginal propensity to spend × real GDP.


833

Figure 16.11 Planned Spending in the Aggregate Expenditure Model

The intercept in Figure 16.11 "Planned Spending in the Aggregate ExpenditureModel" is called autonomous spending. It represents the amount of spending thatthere would be in an economy if income (GDP) were zero. We expect that this willbe positive for two reasons: (1) if a household finds its income is zero, it will stillwant to consume something, so it will either draw on its existing wealth (pastsavings) or borrow against future income; and (2) the government would spendmoney even if GDP were zero.

The slope of the line in Figure 16.11 "Planned Spending in the AggregateExpenditure Model" is given by the marginal propensity to spend. For the reasonsthat we have just explained, we expect that this is positive: increases in income leadto increased spending. However, we expect the marginal propensity to spend to beless than one.

The aggregate expenditure model is based on the two equations we have justdiscussed. We can solve the model either graphically or using algebra. The graphicalapproach relies on Figure 16.12. On the horizontal axis is the level of real GDP. Onthe vertical axis is the level of spending as well as the level of GDP. There are twolines shown. The first is the 45° line, which equates real GDP on the horizontal axis


16.19 The Aggregate Expenditure Model 834

with real GDP on the vertical axis. The second line is the planned spending line. Theintersection of the spending line with the 45° line gives the equilibrium level ofoutput.

Figure 16.12

More Formally

We can also solve the model algebraically. Let us use Y to denote the level of realGDP and E to denote planned expenditure. We represent the marginal propensity tospend by β. The two equations of the model are as follows:

Y = E

and

E = E0 + β × Y.



Here, E0 is autonomous expenditure. We can solve the two equations to find the

values of E and Y that are consistent with both equations. Substituting for E in thefirst equation, we find that

The equilibrium level of output is the product of two terms. The first term—(1/(1 −β))—is called the multiplier. If, as seems reasonable, β lies between zero and one,the multiplier is greater than one. The second term is the autonomous level ofspending.

Here is an example. Suppose that

C = 100 + 0.6Y,

I = 400,

G = 300,

and

NX = 200 − 0.1Y,

where C is consumption, I is investment, G is government purchases, and NX is netexports. First group the components of spending as follows:

C + I + G + NX = (100 + 400 + 300 + 200) + (0.6Y − 0.1Y)

Adding together the first group of terms, we find autonomous spending:

E0 = 100 + 400 + 300 + 200 = 1,000.

Adding the coefficients on the income terms, we find the marginal propensity tospend:

β = 0.6 − 0.1 = 0.5.

Using β = 0.5, we calculate the multiplier:

Y equil = ( 11 − β) × E0 .



We then calculate real GDP:

Y = 2 × 1,000 = 2,000.


• Chapter 7 "The Great Depression"• Chapter 10 "Understanding the Fed"• Chapter 12 "Income Taxes"• Chapter 15 "The Global Financial Crisis"

( 11 − β) = ( 1

1−. 5) = 2.



16.20 Price Adjustment

The price adjustment equation summarizes, at the level of an entire economy, allthe decisions about prices that are made by managers throughout the economy. Theprice adjustment equation is as follows:

inflation rate = autonomous inflation − inflation sensitivity × output gap.

The equation tells us that there are two reasons for rising prices. The first isbecause the output gap is negative. The output gap is the difference betweenpotential output and actual output:

output gap = potential real gross domestic product (real GDP) − actual real GDP.

A positive gap means that the economy is in recession—below potential output. Ifthe economy is in a boom, then the output gap is negative.

The second reason for rising prices is that autonomous inflation is positive.Autonomous inflation refers to the inflation rate that prevails in an economy whenan economy is at potential output (so the output gap is zero). Looking at the secondterm of the price adjustment equation, we see that when real GDP is greater thanpotential output, the output gap is negative, so there is upward pressure on pricesin the economy. The inflation rate will exceed autonomous inflation. By contrast,when real GDP is less than potential output, the output gap is negative, so there isdownward pressure on prices. The inflation rate will be below the autonomousinflation rate. The “inflation sensitivity” tells us how responsive the inflation rate isto the output gap.

The output gap matters because, as GDP increases relative to potential output, laborand other inputs become scarcer. Firms are likely to see rising costs and increasetheir prices as a consequence. Even leaving this aside—that is, even when aneconomy is at potential output—firms are likely to increase their prices somewhat.For example, firms may anticipate that their suppliers or their competitors arelikely to increase prices in the future. A natural response is to increase prices, soautonomous inflation is positive. Figure 16.13 "Price Adjustment" shows the priceadjustment equation graphically.


838

Figure 16.13 Price Adjustment


• Chapter 7 "The Great Depression"• Chapter 10 "Understanding the Fed"• Chapter 11 "Inflations Big and Small"• Chapter 12 "Income Taxes"


16.20 Price Adjustment 839

16.21 Consumption and Saving

The consumption function is a relationship between current disposable income andcurrent consumption. It is intended as a simple description of household behaviorthat captures the idea of consumption smoothing. We typically suppose theconsumption function is upward-sloping but has a slope less than one. So asdisposable income increases, consumption also increases but not as much. Morespecifically, we frequently assume that consumption is related to disposable incomethrough the following relationship:

consumption = autonomous consumption + marginal propensity to consume ×disposable income.

A consumption function of this form implies that individuals divide additionalincome between consumption and saving.

• We assume autonomous consumption is positive. Households consumesomething even if their income is zero. If a household has accumulateda lot of wealth in the past or if a household expects its future income tobe larger, autonomous consumption will be larger. It captures both thepast and the future.

• We assume that the marginal propensity to consume is positive. Themarginal propensity to consume captures the present; it tells us howchanges in current income lead to changes in current consumption.Consumption increases as current income increases, and the larger themarginal propensity to consume, the more sensitive current spendingis to current disposable income. The smaller the marginal propensityto consume, the stronger is the consumption-smoothing effect.

• We also assume that the marginal propensity to consume is less thanone. This says that not all additional income is consumed. When ahousehold receives more income, it consumes some and saves some.

Figure 16.14 "The Consumption Function" shows this relationship.


840

Figure 16.14 The Consumption Function

More Formally

In symbols, we write the consumption function as a relationship betweenconsumption (C) and disposable income (Yd):

C = a + bYd

where a and b are constants. Here a represents autonomous consumption and b isthe marginal propensity to consume. We assume three things about a and b:

1. a > 02. b > 03. b < 1

The first assumption means that even if disposable income is zero (Yd = 0),consumption will still be positive. The second assumption means that the marginalpropensity to consume is positive. By the third assumption, the marginal


16.21 Consumption and Saving 841

propensity to consume is less that one. With 0 < b < 1, part of an extra dollar ofdisposable income is spent.

What happens to the remainder of the increase in disposable income? Sinceconsumption plus saving is equal to disposable income, the increase in disposableincome not consumed is saved. More generally, this link between consumption andsaving (S) means that our model of consumption implies a model of saving as well.

Using

Yd = C + S

and

C = a + bYd

we can solve for S:

S = Yd − C = −a + (1 − b)Yd.

So −a is the level of autonomous saving and (1 − b) is the marginal propensity tosave.

We can also graph the savings function. The savings function has a negativeintercept because when income is zero, the household will dissave. The savingsfunction has a positive slope because the marginal propensity to save is positive.

Economists also often look at the average propensity to consume (APC), whichmeasures how much income goes to consumption on average. It is calculated asfollows:

APC = C/Yd.

When disposable income increases, consumption also increases but by a smalleramount. This means that when disposable income increases, people consume asmaller fraction of their income: the average propensity to consume decreases.Using our notation, we are saying that using C = a + bYd, so we can write

APC = a/Yd + b.



An increase in disposable income reduces the first term, which also reduces theAPC.


• Chapter 12 "Income Taxes"



16.22 The Government Budget Constraint

Like households, governments are subject to budget constraints. These can beviewed in two ways, either within a single year or across many years.

The Single-Year Government Budget Constraint

In any given year, money flows into the government sector, primarily from thetaxes that it imposes on individuals and corporations. We call these governmentrevenues. Money flows out in the form of outlays: government purchases of goodsand services and government transfers. The circular flow of income tells us that anydifference between government purchases and transfers and government revenuesrepresents a government deficit.

Often, we find it useful to group taxes and transfers together as “net taxes” andseparate out government purchases, as in the last line of our definition.

When outflows are less than inflows, then we say a government is running asurplus. In other words, a negative government deficit is the same as a positivegovernment surplus, and a negative government surplus is the same as a positivegovernment deficit.

government surplus = −government deficit.

When a government runs a deficit, it must borrow from the financial markets.When a government runs a surplus, these funds flow into the financial markets andare available for firms to borrow. A government surplus is sometimes calledgovernment saving.

Intertemporal Government Budget Constraint

Tax and spending decisions at different dates are linked. Although governments canborrow or lend in a given year, a government’s total spending over time must be

government deficit ====

outlays − revenuesgovernment purchases + transfers − tax revenuesgovernment purchases − (tax revenues − transfers)government purchases − net taxes.


844

matched with revenues. When a government runs a deficit, it typically borrows tofinance it. It borrows by issuing more government debt (government bonds).

To express the intertemporal budget constraint, we introduce a measure of thedeficit called the primary deficit. The primary deficit is the difference betweengovernment outlays, excluding interest payments on the debt, and governmentrevenues. The primary surplus is minus the primary deficit and is the differencebetween government revenues and government outlays, excluding interestpayments on the debt.

The intertemporal budget constraint says that if a government has some existingdebt, it must run surpluses in the future so that it can ultimately pay off that debt.Specifically, it is the requirement that

current debt outstanding = discounted present value of future primary surpluses.

This condition means that the debt outstanding today must be offset by primarybudget surpluses in the future. Because we are adding together flows in the future,we have to use the tool of discounted present value. If, for example, the currentstock of debt is zero, then the intertemporal budget constraint says that thediscounted present value of future primary surpluses must equal zero.

The stock of debt is linked directly to the government budget deficit. As we notedearlier, when a government runs a budget deficit, it finances the deficit by issuingnew debt. The deficit is a flow that is matched by a change in the stock of governmentdebt:

change in government debt (in given year) = deficit (in given year).

The stock of debt in a given year is equal to the deficit over the previous year plusthe stock of debt from the start of the previous year. If there is a governmentsurplus, then the change in the debt is a negative number, so the debt decreases.The total government debt is simply the accumulation of all the previous years’deficits.

When a government borrows, it must pay interest on its debt. These interestpayments are counted as part of the deficit (they are included in transfers). If agovernment wants to balance the budget, then government spending must actuallybe less than the amount government receives in the form of net taxes (excludinginterest).


16.22 The Government Budget Constraint 845

This presentation of the tool neglects one detail. There is another way in which agovernment can fund its deficit. As well as issuing government debt, it can printmoney. More precisely, then, every year,

change in government debt = deficit − change in money supply.

Written this way, the equation tells us that the part of the deficit that is notfinanced by printing money results in an increase in the government debt.

More Formally

We often denote government purchases of goods and services by G and net taxrevenues (tax revenues minus transfers) by T. The equation for tax revenues is asfollows:

T = τ × Y,

where τ is the tax rate on income and Y is real gross domestic product (real GDP).The deficit is given as follows:

government deficit = G − T = G − τ × Y.

From this equation, the deficit depends on the following:

• Fiscal policy through the choices of G and τ• The level of economic activity (Y)


• Chapter 11 "Inflations Big and Small"• Chapter 12 "Income Taxes"• Chapter 13 "Social Security"• Chapter 14 "Balancing the Budget"• Chapter 15 "The Global Financial Crisis"


16.22 The Government Budget Constraint 846

16.23 The Life-Cycle Model of Consumption

The life-cycle model of consumption looks at the lifetime consumption and savingdecisions of an individual. The choices made about consumption and saving dependon income earned over an individual’s entire lifetime. The model has two keycomponents: the lifetime budget constraint and individual choice given thatconstraint.

Consider the consumption/saving decision of an individual who expects to work fora known number of years and be retired for a known number of years thereafter.Suppose his disposable income is the same in every working year, and he will alsoreceive an annual retirement income—again the same in every year. According tothe life-cycle model of consumption, the individual first calculates the discountedpresent value (DPV) of lifetime income:

DPV of lifetime income = DPV of income from working + DPV of retirement income.

(If the real interest rate is zero, then the DPV calculation simply involves addingincome flows across years.)

We assume the individual wants to consume at the same level in each period of life.This is called consumption smoothing. In the special case of a zero real interestrate, we have the following:

More Formally

Suppose an individual expects to work for a total of N years and to be retired for Ryears. Suppose his disposable income is equal to Yd in every year, and he receivesannual retirement income of Z. Then lifetime income, assuming a zero real interestrate, is given as follows:

lifetime income = NYd + RZ.

If we suppose that he wants to have perfectly smooth consumption, equal to C ineach year, then his total lifetime consumption will be

C × (N + R).

annual consumption =lifetime income

number of years of life.


847

The lifetime budget constraint says that lifetime consumption equals lifetimeincome:

C × (N + R) = NYd + RZ.

To obtain his consumption, we simply divide this equation by the number of yearshe is going to live (N + R):

Provided that income during working years is greater than income in retirementyears, the individual will save during his working years and dissave duringretirement.

If the real interest rate is not equal to zero, then the basic idea is the same—anindividual smooths consumption based on a lifetime budget constraint—but thecalculations are more complicated. Specifically, the lifetime budget constraint mustbe written in terms of the discounted present values of income and consumption.


• Chapter 7 "The Great Depression"• Chapter 13 "Social Security"• Chapter 14 "Balancing the Budget"

C =NY d + RZ

N + R.


16.23 The Life-Cycle Model of Consumption 848

16.24 Aggregate Supply and Aggregate Demand

The aggregate supply and aggregate demand (ASAD) model is presented here. Tounderstand the ASAD model, we need to explain both aggregate demand andaggregate supply and then the determination of prices and output.

The aggregate demand curve tells us the level of expenditure in an economy for agiven price level. It has a negative slope: the demand for real gross domesticproduct (real GDP) decreases when the price level increases. The downward slopingaggregate demand curve does not follow from the microeconomic “law of demand.”As the price level increases, all prices in an economy increase together. Thesubstitution of expensive goods for cheap goods, which underlies the law ofdemand, does not occur in the aggregate economy.

Instead, the downward sloping demand curve comes from other forces. First, asprices rise, the real value of nominal wealth falls, and this leads to a fall inhousehold spending. Second, as prices rise today relative to future prices,households are induced to postpone consumption. Finally, a higher price level canlead to a higher interest rate through the response of monetary policy. All thesefactors together imply that higher prices lead to lower overall demand for real GDP.

Aggregate supply is equal to potential output at all prices. Potential output isdetermined by the available technology, physical capital, and labor force and isunaffected by the price level. Thus the aggregate supply curve is vertical. Incontrast to a firm’s supply curve, as the price level increases, all prices in aneconomy increase. This includes the prices of inputs, such as labor, into theproduction process. Since no relative prices change when the price level increases,firms are not induced to change the quantity they supply. Thus aggregate supply isvertical.

The determination of prices and output depends on the horizon: the long run or theshort run. In the long run, real GDP equals potential GDP, and real GDP also equalsaggregate expenditure. This means that, in the long run, the price level must be atthe point where aggregate demand and aggregate supply meet. This is shown inFigure 16.15 "Aggregate Supply and Aggregate Demand in the Long Run".


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Figure 16.15 Aggregate Supply and Aggregate Demand in the Long Run

In the short run, output is determined by aggregate demand at the existing pricelevel. Prices need not be at their long-run equilibrium levels. If they are not, thenoutput will not equal potential output. This is shown in Figure 16.16 "AggregateSupply and Aggregate Demand in the Short Run".


16.24 Aggregate Supply and Aggregate Demand 850

Figure 16.16 Aggregate Supply and Aggregate Demand in the Short Run

The short-run price level is indicated on the vertical axis. The level of output isdetermined by aggregate demand at that price level. As prices are greater than thelong-run equilibrium level of prices, output is below potential output. The pricelevel adjusts over time to its long-run level, according to the price-adjustmentequation.


We do not explicitly use this tool in our chapter presentations. However, the toolcan be used to support the discussions in the following chapters.

• Chapter 7 "The Great Depression"• Chapter 10 "Understanding the Fed"• Chapter 11 "Inflations Big and Small"• Chapter 12 "Income Taxes"


16.24 Aggregate Supply and Aggregate Demand 851

16.25 The IS-LM Model

The IS-LM model provides another way of looking at the determination of the levelof short-run real gross domestic product (real GDP) in the economy. Like theaggregate expenditure model, it takes the price level as fixed. But whereas thatmodel takes the interest rate as exogenous—specifically, a change in the interestrate results in a change in autonomous spending—the IS-LM model treats theinterest rate as an endogenous variable.

The basis of the IS-LM model is an analysis of the money market and an analysis ofthe goods market, which together determine the equilibrium levels of interest ratesand output in the economy, given prices. The model finds combinations of interestrates and output (GDP) such that the money market is in equilibrium. This createsthe LM curve. The model also finds combinations of interest rates and output suchthat the goods market is in equilibrium. This creates the IS curve. The equilibriumis the interest rate and output combination that is on both the IS and the LM curves.

LM Curve

The LM curve represents the combinations of the interest rate and income suchthat money supply and money demand are equal. The demand for money comesfrom households, firms, and governments that use money as a means of exchangeand a store of value. The law of demand holds: as the interest rate increases, thequantity of money demanded decreases because the interest rate represents anopportunity cost of holding money. When interest rates are higher, in other words,money is less effective as a store of value.

Money demand increases when output rises because money also serves as a mediumof exchange. When output is larger, people have more income and so want to holdmore money for their transactions.

The supply of money is chosen by the monetary authority and is independent of theinterest rate. Thus it is drawn as a vertical line. The equilibrium in the moneymarket is shown in Figure 16.17 "Money Market Equilibrium". When the moneysupply is chosen by the monetary authority, the interest rate is the price thatbrings the market into equilibrium. Sometimes, in some countries, central bankstarget the money supply. Alternatively, central banks may choose to target theinterest rate. (This was the case we considered in Chapter 10 "Understanding theFed".) Figure 16.17 "Money Market Equilibrium" applies in either case: if the


852

monetary authority targets the interest rate, then the money market tells us whatthe level of the money supply must be.

Figure 16.17 Money Market Equilibrium

To trace out the LM curve, we look at what happens to the interest rate when thelevel of output in the economy changes and the supply of money is held fixed.Figure 16.18 "A Change in Income" shows the money market equilibrium at twodifferent levels of real GDP. At the higher level of income, money demand is shiftedto the right; the interest rate increases to ensure that money demand equals moneysupply. Thus the LM curve is upward sloping: higher real GDP is associated withhigher interest rates. At each point along the LM curve, money supply equalsmoney demand.

We have not yet been specific about whether we are talking about nominal interestrates or real interest rates. In fact, it is the nominal interest rate that represents theopportunity cost of holding money. When we draw the LM curve, however, we putthe real interest rate on the axis, as shown in Figure 16.19 "The LM Curve". Thesimplest way to think about this is to suppose that we are considering an economywhere the inflation rate is zero. In this case, by the Fisher equation, the nominaland real interest rates are the same. In a more complete analysis, we can


16.25 The IS-LM Model 853

incorporate inflation by noting that changes in the inflation rate will shift the LMcurve. Changes in the money supply also shift the LM curve.

Figure 16.18 A Change in Income



Figure 16.19 The LM Curve

IS Curve

The IS curve relates the level of real GDP and the real interest rate. It incorporatesboth the dependence of spending on the real interest rate and the fact that, in theshort run, real GDP equals spending. The IS curve is shown in Figure 16.18 "AChange in Income". We label the horizontal axis “real GDP” since, in the short run,real GDP is determined by aggregate spending. The IS curve is downward sloping: asthe real interest rate increases, the level of spending decreases.



Figure 16.20 The IS Curve

In fact, we derived the IS curve in Chapter 10 "Understanding the Fed". Thedependence of spending on real interest rates comes partly from investment. As thereal interest rate increases, spending by firms on new capital and spending byhouseholds on new housing decreases. Consumption also depends on the realinterest rate: spending by households on durable goods decreases as the realinterest rate increases.

The connection between spending and real GDP comes from the aggregateexpenditure model. Given a particular level of the interest rate, the aggregateexpenditure model determines the level of real GDP. Now suppose the interest rateincreases. This reduces those components of spending that depend on the interestrate. In the aggregate expenditure framework, this is a reduction in autonomousspending. The equilibrium level of output decreases. Thus the IS curve slopesdownwards: higher interest rates are associated with lower real GDP.

Equilibrium

Combining the discussion of the LM and the IS curves will generate equilibriumlevels of interest rates and output. Note that both relationships are combinations of



interest rates and output. Solving these two equations jointly determines theequilibrium. This is shown graphically in Figure 16.21. This just combines the LMcurve from Figure 16.19 "The LM Curve" and the IS curve from Figure 16.20 "The ISCurve". The crossing of these two curves is the combination of the interest rate andreal GDP, denoted (r*,Y*), such that both the money market and the goods marketare in equilibrium.

Figure 16.21

Equilibrium in the IS-LM Model.

Comparative Statics

Comparative statics results for this model illustrate how changes in exogenousfactors influence the equilibrium levels of interest rates and output. For this model,there are two key exogenous factors: the level of autonomous spending (excludingany spending affected by interest rates) and the real money supply. We can studyhow changes in these factors influence the equilibrium levels of output and interestrates both graphically and algebraically.



Variations in the level of autonomous spending will lead to a shift in the IS curve, asshown in Figure 16.22 "A Shift in the IS Curve". If autonomous spending increases,then the IS curve shifts out. The output level of the economy will increase. Interestrates rise as we move along the LM curve, ensuring money market equilibrium. Onesource of variations in autonomous spending is fiscal policy. Autonomous spendingincludes government spending (G). Thus an increase in G leads to an increase inoutput and interest rates as shown in Figure 16.22 "A Shift in the IS Curve".

Figure 16.22 A Shift in the IS Curve

Variations in the real money supply shift the LM curve, as shown in Figure 16.23 "AShift in the LM Curve". If the money supply decreases, then the LM curve shifts in.This leads to a higher real interest rate and lower output as the LM curve shiftsalong the fixed IS curve.



Figure 16.23 A Shift in the LM Curve

More Formally

We can represent the LM and IS curves algebraically.

LM Curve

Let L(Y,r) represent real money demand at a level of real GDP of Y and a realinterest rate of r. (When we say “real” money demand, we mean that, as usual, wehave deflated by the price level.) For simplicity, suppose that the inflation rate iszero, so the real interest rate is the opportunity cost of holding money.If we wantedto include inflation in our analysis, we could write the real demand for money asL(Y, r + π), where π is the inflation rate. Assume that real money demand takes aparticular form:

L(Y,r) = L0 + L1Y – L2r.

In this equation, L0, L1, and L2 are all positive constants. Real money demand is

increasing in income and decreasing in the interest rate. Letting M/P be the realstock of money in the economy, then money market equilibrium requires



M/P = L0 + L1Y – L2r.

Given a level of real GDP and the real stock of money, this equation can be used tosolve for the interest rate such that money supply and money demand are equal.This is given by

r = (1/L2) [L0 + L1Y – M/P].

From this equation we learn that an increase in the real stock of money lowers theinterest rate, given the level of real GDP. Further, an increase in the level of realGDP increases the interest rate, given the stock of money. This is another way ofsaying that the LM curve is upward sloping.

IS Curve

Recall the two equations from the aggregate expenditure model:

Y = E

and

E = E0(r) + βY.

Here we have shown explicitly that the level of autonomous spending depends onthe real interest rate r.

We can solve the two equations to find the values of E and Y that are consistent withboth equations. We find

Given a level of the real interest rate, we solve for the level of autonomous spending(using the dependence of consumption and investment on the real interest rate)and then use this equation to find the level of output.

Here is an example. Suppose that

C = 100 + 0.6Y,

I = 400 − 5r,

Y equil = ( 11 − β) × E0 (r).



G = 300,

and

NX = 200 − 0.1Y,

where C is consumption, I is investment, G is government purchases, and NX is netexports. First group the components of spending as follows:

C + I + G + NX = (100 + 400 − 5r + 300 + 200) + (0.6Y − 0.1Y)

Adding together the first group of terms, we find autonomous spending:

E0 = 100 + 400 + 300 + 200 − 5r = 1000 − 5r.

Adding the coefficients on the income terms, we find the marginal propensity tospend:

β = 0.6 − 0.1 = 0.5.

Using β = 0.5, we calculate the multiplier:

We then calculate real GDP, given the real interest rate:

Y = 2 × (1000 − 5r) = 2000 − 10r.

Equilibrium

Combining the discussion of the LM and the IS curves will generate equilibriumlevels of interest rates and output. Note that both relationships are combinations ofinterest rates and output. Solving these two equations jointly determines theequilibrium.

Algebraically, we have an equation for the LM curve:

r = (1/L2) [L0 + L1Y – M/P].

And we have an equation for the IS curve:

( 11 − β) = ( 1

1−. 5) = 2.



Y = mE0(r),

where we let m = (1/(1 – β)) denote the multiplier. If we assume that thedependence of spending in the interest rate is linear, so that E0(r) = e0 – e1r, then

the equation for the IS curve is

Y = m (e0-e1r),

To solve the IS and LM curves simultaneously, we substitute Y from the IS curveinto the LM curve to get

r = (1/L2) [L0 + L1 m(e0-e1r) – M/P].

Solving this for r we get

r = Ar – BrM/P.

where both Ar and Br are constants, with Ar = (L0 + L1me0)/(L1me1 + L2) and Br =

1/(L1me1 + L2). This equation gives us the equilibrium level of the real interest rate

given the level of autonomous spending, summarized by e0, and the real stock of

money, summarized by M/P.

To find the equilibrium level of output, we substitute this equation for r back intothe equation for the IS curve. This gives us

Y = Ay + By(M/P),

where both Ay and By are constants, with Ay = m(e0 – e1Ar) and By = me1Br. This

equation gives us the equilibrium level of output given the level of autonomousspending, summarized by e0, and the real stock of money, summarized by M/P.

Algebraically, we can use the equations to determine the magnitude of theresponses of interest rates and output to exogenous changes. An increase in theautonomous spending, e0, will increase both Ar and Ay, implying that both the

interest rate and output increase.To see that Ay increases with e0 requires a bit

more algebra. An increase in the real money stock will reduce interest rates by Br

and increase output by By. A key part of monetary policy is the sensitivity of

spending to the interest rate, given by e1. The more sensitive is spending to the

interest rate, the larger is e1 and therefore the larger is By.




We do not explicitly use this tool in our chapter presentations. However, the toolcan be used to support the discussions in the following chapters.

• Chapter 9 "Money: A User’s Guide"• Chapter 10 "Understanding the Fed"• Chapter 11 "Inflations Big and Small"• Chapter 14 "Balancing the Budget"



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This is “Macroeconomics Toolkit”, chapter 16 from the ... · Chapter 16 Macroeconomics Toolkit. In this chapter, we present the key tools used in the macroeconomics part of this